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Statistical Mechanics,Kinetic Theory
Ideal Gas
8.01tNov 22, 2004
Statistical Mechanics and Thermodynamics
• Thermodynamics Old & Fundamental – Understanding of Heat (I.e. Steam) Engines – Part of Physics Einstein held inviolate – Relevant to Energy Crisis of Today
• Statistical Mechanics is Modern Justification –Based on mechanics: Energy, Work, Momentum –Ideal Gas model gives observed thermodynamics
• Bridging Ideas Temperature (at Equilibrium) is Average Energy Equipartition - as simple/democratic as possible
Temperature and Equilibrium• Temperature is Energy per Degree of Freedom
– More on this later (Equipartition)
• Heat flows from hotter to colder object Until temperatures are equal Faster if better thermal contact Even flows at negligible ∆t (for reversible process)
• The Unit of Temperature is the Kelvin Absolute zero (no energy) is at 0.0 K Ice melts at 273.15 Kelvin (0.0 C) Fahrenheit scale is arbitrary
State Variables of System
• State Variables - Definition Measurable Static Properties Fully Characterize System (if constituents known) e.g. Determine Internal Energy, compressibilityRelated by Equation of State
• State Variables: Measurable Static Properties– Temperature - measure with thermometer – Volume (size of container or of liquid in it) – Pressure (use pressure gauge) – Quantity: Mass or Moles or Number of Molecules
• Of each constituent or phase (e.g. water and ice)
Equation of State• A condition that the system must obey
– Relationship among state variables
• Example: Perfect Gas Law – Found in 18th Century Experimentally
– pV = NkT = nRT
– K is Boltzmann’s Constant 1.38x10-23 J/K – R is gas constant 8.315 J/mole/K
• Another Eq. Of State is van der Waals Eq.– You don’t have to know this.
PV = n R T = N k T• P is the Absolute pressure
– Measured from Vacuum = 0 – Gauge Pressure = Vacuum - Atmospheric – Atmospheric = 14.7 lbs/sq in = 105 N/m
• V is the volume of the system in m3
– often the system is in cylinder with piston
– Force on the piston does work on world
Does work on world
P=0 outsideT,P,V,n
PV = n R T = N k Tchemists vs physicists
• Mole View (more Chemical) = nRT– R is gas constant 8.315 J/mole/K
• Molecular View (physicists) = NkT– N is number of molecules in system – K is Boltzmann’s Constant 1.38x10-23 J/K
Using PV=nRT
• Recognize: it relates state variables of a gas
• Typical Problems – Lift of hot air balloon – Pressure change in heated can of tomato soup – Often part of work integral
Heat and Work are Processes
• Processes accompany/cause state changes – Work along particular path to state B from A – Heat added along path to B from A
• Processes are not state variables– Processes change the state! – But Eq. Of State generally obeyed
Ideal Gas Law Derivation: Assumptions
• Gas molecules are hard spheres without internal structure
• Molecules move randomly
• All collisions are elastic • Collisions with the wall are elastic and
instantaneous
Gas Properties
• N number of atoms in volume
• nm moles in volume
• m is atomic mass (12C = 12)
nm n N Am m• mass density ρ = mT = = V V V
• Avogadro’s Number N A = 6.02 × 1023 molecules ⋅ mole−1
Motion of Molecules • Assume all molecules have
the same velocity (we will drop this latter)
• The velocity distribution is isotropic
Collision with Wall
p• Change of i : ∆ = −mvx f −mv x , x,0
momentum j : ∆ = mv ,py y f −mvy ,0
• Elastic collision vy f = vy ,0 ,
vx ≡ vx, f = vx,0
• Conclusion i : ∆ = −2mvpx x
Momentum Flow Tube
• Consider a tube of cross sectional area A and length v t∆x
• In time ∆t half the molecules in tube hit wall
ρ ρ • Mass enclosed that ∆m = Volume = Av ∆txhit wall 2 2
Pressure on the wall
• Newton’s Second Law G ∆p G − ∆mvx i = − 2ρAv ∆t 2 ixFwall gas = =
2 2
i = −ρAvx, ∆t ∆t 2∆t G G
2 i , ,• Third Law Fgas wall = −Fwall gas = ρAvx
G F
,gas wall 2 • Pressure Ppressure =
A = ρvx
Average velocity
• Replace the square of the velocity with the average of the square of the velocity (v )2
x ave
• random motions imply
2 2 2 2 2( ) = vv ( ) + (v ) + (v ) = 3(v )z xave x ave y ave ave ave
• Pressure 1 2 2 nmN A 1 ( )2P = ρ v m vpressure ave ave3 ( ) =
3 V 2
Degrees of Freedom in Motion
• Three types of degrees of freedom for molecule
1. Translational2. Rotational 3. Vibrational
• Ideal gas Assumption: only 3 translational degrees of freedom are present for moleculewith no internal structure
Equipartition theorem: Kinetic energy and temperature
• Equipartition of Energy Theorem
( ) = (#degrees of freedom) 31 m v2 kT = kT ave 2 2 2
−23 −1• Boltzmann Constant k = 1.38×10 J ⋅ K
• Average kinetic of gas molecule defines kinetic temperature
Ideal Gas Law
• Pressure
2 nmN A 1 2 nmN A kT= m v2 nmN A 3
kT = ave
ppressure 3 V 2 ( ) = 3 V 2 V
• Avogadro’s Number N A = 6.022 ×1023 molecules ⋅mole−
1• Gas Constant R = N Ak = 8.31 J ⋅mole−1 ⋅ K −
• Ideal Gas Law pV = n RT m
1
Ideal Gas Atmosphere
• Equation of State PV n RT = m
• Let Mmolar be the molar mass.
• Mass Density ρ = nm Mmolar =
Mmolar PV RT
• Newton’s Second Law
A P z (( ( )− P z + ∆z))− ρgA∆z = 0
Isothermal Atmosphere
• Pressure Equation P z + ∆z )− P z( ( )= ρg
∆z
• Differential equation dP = −ρg = −
Mmolar g P
dz RT P z z • Integration ( )dp
= − ∫ Mmolar g
dz′ P∫ 0
p RT z =0
• Solution ⎛ P z( )⎞⎟ = −
Mmolar gln ⎜ zRT⎝ P0 ⎠
• Exponentiate ( )= P0 exp
⎛−
Mmolar g z ⎞
P z ⎜⎝ RT ⎠⎟